# Non-Forced-Choice Models (feasible outside option)¶

## Utility Maximization with an Outside Option¶

### Strict¶

A general choice dataset \(\mathcal{D}\) on a set of alternatives \(X\) is explained by
**(strict) utility maximization with an outside option** if
there is a strict linear order \(\succ\) on \(X\) and an alternative \(x^*\in X\) such
that for every menu \(A\) in \(\mathcal{D}\)

where

is the strictly most preferred alternative in \(A\) according to \(\succ\).

### Non-strict¶

A general choice dataset \(\mathcal{D}\) on a set of alternatives \(X\) is explained by
**(non-strict) utility maximization with an outside option** if
there is a weak order \(\succsim\) on \(X\) and an alternative \(x^*\in X\) such
that for every menu \(A\) in \(\mathcal{D}\)

**and**

where

is the set of weakly most preferred alternatives in \(A\) according to \(\succsim\).

## Overload-Constrained Utility Maximization¶

### Strict¶

A general choice dataset \(\mathcal{D}\) on a set of alternatives \(X\) is explained by
**(strict) overload-constrained utility maximization** if there is a strict linear order
\(\succ\) on \(X\) and an integer \(n\) such that for every menu \(A\) in \(\mathcal{D}\)

where

is the strictly most preferred alternative in \(A\) according to \(\succ\).

### Non-strict¶

A general choice dataset \(\mathcal{D}\) on a set of alternatives \(X\) is explained by
**(non-strict) overload-constrained utility maximization** if there is a weak order
\(\succsim\) on \(X\) and an integer \(n\) such that for every menu \(A\) in \(\mathcal{D}\)

**and**

where

is the set of weakly most preferred alternatives in \(A\) according to \(\succsim\).

## Incomplete-Preference Maximization: Maximally Dominant Choice¶

### Strict¶

A general choice dataset on a set of alternatives \(X\) is explained by
**(strict) maximally dominant choice** if there is a strict partial order
\(\succ\) on \(X\) such that for every menu \(A\) in \(\mathcal{D}\)

where

is the (possibly non-existing) strictly most preferred alternative in \(A\) according to \(\succ\).

### Non-strict¶

A general choice dataset \(\mathcal{D}\) on a set of alternatives \(X\) is explained by
**(non-strict) maximally dominant choice** if there is an incomplete preorder
\(\succsim\) on \(X\) such that for every menu \(A\) in \(\mathcal{D}\)

**and**

where

is the (possibly empty) set of the weakly most preferred alternatives in \(A\) according to \(\succsim\).

## Incomplete-Preference Maximization: Partially Dominant Choice (non-forced)¶

A general choice dataset \(\mathcal{D}\) on a set of alternatives \(X\) is explained by
**partially dominant choice (non-forced)** if there exists a strict partial order \(\succ\) on \(X\)
such that for every menu \(A\) in \(\mathcal{D}\) with at least two alternatives

Note

In its distance-score computation of this model, Prest penalizes deferral/choice of the outside option at singleton menus. Although this is not a formal requirement of the model, its predictions at non-singleton menus are compatible with the assumption that all alternatives are desirable, and hence that active choices be made at all singletons.